Expander graphs, gonality, and variation of Galois representations

By Jordan S. Ellenberg, Chris Hall, Emmanuel Kowalski. 2010

50 citations recorded by the source corpus.

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in one-parameter families of abelian varieties.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eda1bac2fce6f5e305b55a3aed37066ce60c6da4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Peter Sarnak
2. P. P. Varj'u
3. Alexander Lubotzky
4. László Pyber
5. E. Szab'o
6. P. Habegger
7. V. Blomer
8. Gergely Harcos

In Computational Theory and Mathematics

1. P. P. Varj'u
2. László Pyber
3. Peter Sarnak
4. Alexander Lubotzky
5. E. Szab'o
6. P. Habegger

In Geometry and Topology

1. Peter Sarnak
2. P. P. Varj'u
3. László Pyber
4. Alexander Lubotzky
5. E. Szab'o
6. P. Habegger
7. Alex Kontorovich
8. Emmanuel Breuillard

Citations in the corpus

• Growth in finite simple groups of Lie type of bounded rank (2010, methodology): “For an unexpected application of Theorem 5 in arithmetic geometry, see [38].”
• Notes on Thin Matrix Groups (2012, background): “Recently Ellenberg, Hall and Kowalski [Ellenberg et al. 2012] have applied similar reasoning to a diophantine problem on a tower of curves.”
• Nilprogressions and groups with moderate growth (2015, unknown): “n ˆ 2−2, |S| |G| ε˙ . Then G is ε/3-almost flat. In particular, the conclusion of Theorem 1.3 holds. This is useful in a variety of contexts, for example in the work of Ellenberg, Hall and Kowalski [19] in arithmetic geometry, where they used a weak lower bound on λ1 for certain perfect groups due t”
• Expander graphs in pure and applied mathematics (2011, methodology): “atment with stronger results is given in [EMV]. The modern approach makes use of random walks on expander graphs. (VI) Counting rational solutions on curves: Expanders are used in a surprising way in [EHK] to show some strong niteness results on the number of k-rational points on some families of cu”
• Random walks in compact groups (2012, background): “We assume further that for Γ ∈ Ω2, the quotient G/Γ has poly-log spectral gap (i.e. (4) is satisfied).”
• Small height and infinite nonabelian extensions (2011, methodology): “gree of the morphism to the base of the tower. This corresponds precisely to the Bogomolov property transposed to extensions of a fixed function field. In more recent work Ellenberg, Hall, and Kowalski [15] deduced lower bounds for the gonality in more general towers of curves. Poonen provided good ev”